Measures of central tendency

CFA level I / Quantitative Methods: Basic Concepts / Statistical Concepts and Market Returns / Measures of central tendency

The measures of central tendency specify where the data are centered. These are more widely used than any other statistical measure because they can be computed easily. The measures of location include not only measures of central tendency (mean, median, and mode) but other measures (quartiles, quintiles, deciles, and percentiles) that illustrate the location or distribution of data.

The arithmetic mean is the sum of the observations divided by the number of observations.

The population mean (µ) is the arithmetic mean value of a population whereas a sample mean (X-bar) is the arithmetic mean value of a sample.

µ = ∑X_i/N (i from 1 to N)

X-bar = ∑X_i/n (i from 1 to n)

where
X_i = ith observation
n = number of observations in the sample
N = the number of observations in the population

The mean of observations of some units at a specific point in time is called as cross-sectional mean. The mean of observation of the same unit over a period of time is called as time-series mean.

Properties of the arithmetic mean: The sum of deviations from the mean for all the observations is zero. The deviation from the mean indicates risk.

The arithmetic mean multiplied by the number of observations gives us the sum of all observations.

The mean uses all the information about the size and magnitude of the observations, unlike median and mode.

The drawback of arithmetic mean is its sensitivity to the extreme values. The most common approach to handling the problem of extreme values is to report median instead of mean.

Example 3: Arithmetic mean

The percentage annual return by a fund manager over the years in percentage are 3, 5, 7, -6, -5, 6, 2, 0, 5, -8, -4, 4, 184, 4, 6, 1. Compute the population arithmetic mean. Also, calculate the sample mean considering the first ten observations. What is the problem with this population arithmetic mean? Solution: Population arithmetic mean = (3+5+7-6-5+6+2+0+5-8-4+4+184+4+6+1)/16 = 204/16 = 12.75 percent. Sample arithmetic mean = (3+5+7-6-5+6+2+0+5-8)/10 = 9/10 = 0.9 percent. The problem with the population arithmetic mean is that it has been affected by an extreme value of 184 percent return. Because of that the average return has come out to be 12.75 percent and the fund has achieved return more than 7 percent only once and still, the average is so high.

The weighted mean is slightly different than the arithmetic mean as it gives different weights (probabilities) to different outcomes. It is extremely important in revenue forecasting because the futures events can have different probabilities. For example, the growth rate in revenue can depend on the policies formed by the government. If the policies are favorable, then the revenue would grow by 15 percent, and if the policies are unfavorable, then the revenue would grow by 5 percent. Now, the probability of favorable policies can be different than the probability of unfavorable policies. Suppose the probability of favorable policies is 80 percent and that of unfavorable policies is 20 percent. Then the weighted mean (expected value) of revenue growth rate will be 0.8*0.15 + 0.2*0.05 = 13 percent. The arithmetic mean would have given a value of 10 percent by giving equal weight to both the events.

The weighted mean is also extremely important in the calculation of portfolio return when the different percentage of money is invested in different assets.

Weighted mean = X_w-bar = ∑w_iX_i (i from 1 to N)
where ∑w_i = 1 (i from 1 to N)

w_i = weight of ith observation
X_i = value of ith observation

Example 4: Weighted average mean

The constituents of the portfolio, the weights and the expected return of the securities are given in the table below. Compute the expected return from the portfolio. Securities Weight Expected Return Security A 0.30 0.08 Security B 0.10 0.04 Security C 0.05 0.14 Security D 0.15 0.09 Security E 0.40 0.10 Solution: The expected return from the portfolio would be equal to the weighted average mean. The weighted mean = 0.30*0.08 + 0.10*0.04 + 0.05*0.14 + 0.15*0.09 + 0.40*0.10 = 0.0885 = 8.85 percent.

The geometric mean is most frequently used to average a time series of rates of return on an asset. The geometric mean makes the most sense when comparing the returns of portfolio manager over a time horizon. It is calculated by using the following formula:

Geometric mean (GM) = (X₁X₂X₃.....X_n)^(1/n)
with X_i ≥ 0 for i =1,2,3,...,n.

Taking log on both sides of the equation, we get:

ln(GM) = (1/n)*ln(X₁X₂X₃.....X_n) = (1/n)∑lnX_i i from 1 to n

For calculating the geometric mean of the returns over the years, we need to add one to each return because returns can be negative as well. But the maximum negative return could be minus 100 percent. So, on adding 1 to the return, the number will never be negative. The formula in such case is given below:

GM = [(1+r₁)(1+r₂)(1+r₃)........(1+r_n)]^(1/n) - 1

Example 5: Arithmetic mean and Geometric mean

The market value of a mutual fund at the beginning of last five years are $5.0 million, $15.0 million, $10 million, $5 million and $2.5 million respectively. The fund value is $4.0 million today. No external money has been withdrawn from or added to the fund. Compute the geometric mean and arithmetic mean of return over the last five years and interpret the result. Solution: Let's first calculate the return earned by the fund each year. Year Value of fund at the beginning Value of fund at the end Return (in percent) 1 5 15 200.00 2 15 10 -33.33 3 10 5 -50.00 4 5 2.5 -50.00 5 2.5 4 60.00 Arithmetic mean = (200 - 33.33 - 50 - 50 + 60)/5 = 25.33 percent Geometric mean = [(1+2)(1-0.33)(1-0.5)(1-0.5)(1+0.6)](1/5) - 1 = -4.36 percent The geometric mean is also known as compounded annual growth rate (CAGR) and can be calculated as given below: CAGR = (Value at the end of five year/Value at the beginning of first year)(1/5) - 1 = (4/5)(1/5) - 1 = -4.36 percent The above formula of CAGR is valid only when there are no interim cash flows. We can see the problem with the arithmetic mean here. The value of the fund has decreased over five years but the arithmetic mean is suggesting that the portfolio has earned 25.33 percent each year. The arithmetic mean is affected by the extreme values is always greater than the geometric mean with different returns. The geometric mean gives the correct return that has been compounded annually, also known as CAGR.

The harmonic mean is used in a limited number of applications such as cost averaging. The weight of observation is inversely proportional to its magnitude in the harmonic mean. Therefore, it is appropriate when averaging ratio when the ratios are repeatedly applied to a fixed quantity to yield a number of variable units. In the case of systematic investment plans (SIPs) for index funds, the fund manager buys the units for a certain amount of money at a certain date every month. Thus, the units of index fund received are different every month, and the number of units is inversely proportional to the index value. When the market falls, you get more units of the index fund, and the cost is averaged down. When the markets are volatile which is usually the case with the equity markets, then this strategy yield the best results and minimize our costs as the harmonic mean is the lowest of all means. That's why this dollar cost averaging is even recommend by Benjamin Graham for uninformed retail investors in his masterpiece The Intelligent Investor.

Harmonic mean = n/∑(1/X_i)
with X_i >0 for i = 1, 2, 3, 4,.., n

The median is the value of the middle observation of a set of observations that have been sorted in an ascending or descending order. When there are an odd number of observations, the median equals the value of observation placed at n(n+1)/2 position. For an even-numbered sample, the median equal the mean of the values of the observations placed at n/2 and (n/2)+1 position. Its advantage over mean is that it is not affected by the extreme values.

Example 6: Calculating median

The P/E ratios of six companies in the pharmaceutical sector are given as 17.5, 12.0, 11.8, 14.6, 159.7, and 13.8. What is the median and mean P/E of the pharmaceutical sector? Solution: Arithmetic mean = (17.5+12.0+11.8+14.6+159.7+13.8)/6 = 38.23 For calculating the median, first arrange the P/E ratios in ascending order: 11.8, 12.0, 13.8, 14.6, 17.5, 159.7 Since the number of observations is an even number, the median will be the arithmetic mean of the observations at (6/2) and (6/2)+1 positions. Observation at 3rd position = 13.8 Observation at 4th position = 14.6 Median = (13.8+14.6)/2 = 14.2 The median is an appropriate P/E value here because most of the companies have comparable P/E as compared to the median value. One extreme value greatly impacts the mean P/E value. A company can have an extreme P/E if its earnings are suppressed due to some particular reason in a year.

The mode is the most frequently occurring value in distribution. A distribution can have one mode, more than one modes, or even no modes. When a distribution has exactly one most frequently item occurring more than once, then that distribution has one mode and is also called as unimodal distribution. If a distribution has more than one item that occurs most frequently, then it is called as multimodal distribution. When all the observations in the distribution are different, then the distribution is said to have no mode.

We can also put the returns of stocks in different intervals and the interval having the maximum number of observations is called modal interval.

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